I think the best solution is using the MaximizeFlash plugin for Chrome. It detects any flash in the page and you can make them fullscreen. To get rid of toolbar up just hit f11. You dont need to install any other apps.

Which creates a small wrapper library that can be pre-loaded before launching firefox or chrome and it fixes the information about the screen sizes that flash uses, so you get full screen flash on either monitor at the correct full size. It is quite simple to set up

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hexafractionOct 7 '12 at 22:03

There is a very similar issue which does not answer exactly your question, but still might be helpful for your usecase. When watching fullscren HTML5 videos (e.g. youtube), the video closes by default if you click anywhere. This behavour can be disabled in Firefox by setting full-screen-api.exit-on-deactivate to false in about:config. Anyhow, this does not effect Flash videos.

If you want fash to go full screen mod, drag the screen you want it to be in, on the left in "screen display" menu. Just switch the positions of the display icons in this menu. this should work with any browser